Mechanics of MaterialsPart 2: Tension, Bending, and Buckling

2

How strong is spaghetti?

3

Estimate the strength of spaghetti

► Estimate how much force can one strand of spaghetti can withstand.

► What happens when you put multiple strands together?

4

Spaghetti can be pretty strong

► The winning bridge from EI at California State University – Fullerton in 2014.

Bridge Weight: 242 g

Bridge Height: 25 cm

Maximum Load:

62.5 kg (138 lbs)!

5

Strength depends on how you load the sample

► We sometimes refer to a force as a “load”. We say that we are loading the

spaghetti when we apply a force to it.

► Make a guess: Spaghetti is strongest when you load it in…

tension / bending / compression

► What happens to the spaghetti when you try to compress it along its length?

6

Compression: crushing and buckling

► What happens to the spaghetti when you try to compress it along its length?

All images are in the public domain.

If the sample is short and

squat, it will be crushed.

If the sample is long and

slender, it will buckle

(bend out to the side).

77

Quiz

► Identify tension, bending, or compression in the photos below.

All images are in the public domain

8

Strength in tension

9

How do you measure the strength of a material in tension?

► The maximum stress a material can handle is the ultimate tensile strength

► To avoid plastic deformation, keep the stress below the yield strength

Graphs from the video https://youtu.be/D8U4G5kcpcM?t=17s; Accessed 6/23/17

This is the yield point where

the deformation changes from

elastic to plastic.

𝝈

𝝐

Yield

strength Point of maximum stress

Ultimate

tensile

strength

10

Failure in tension

► And remember that

1111

Quiz

► The ultimate tensile strength of pine is 40 MPa (5800 psi).

What forces are required to break the samples shown below?

1.5 in

4.0 in

36 in

0.75 in

2.0 in

18 in

The material strength is the same,

but which sample is stronger?

How much stronger?

1212

Quiz

► Order from highest to lowest maximum tensile force.

Area A

Length

L

L

L

L/2

A/2

A

A/4

A B

C

D

13

Strength in bending

14

Factors that affect bending

► What changes could you make so the beam shown below bends less?

► Some ideas:

► Reduce the load (the person’s weight)

► Bring the supports closer together

► Make the beam out of a stronger material

► Make the beam thicker

All images are in the public domain.

1515

Activity: bend a ruler or popsicle stick

► Take a ruler or popsicle stick and try bending

it in the two ways shown below.

► Which way is easier to bend?

All images are in the public domain.

1. Bend so the top face

(in white) is folding in

on itself.

2. Rotate 90 degrees

and try again.

1616

Why does the bending direction make a difference?

► When a sample is loaded by bending, the shape and orientation of the cross-

section makes a difference.

► The two diagrams shown below have the same cross-sectional area, but the

one on the right has a higher area moment of inertia, which measures

resistance to bending.

► Area moment of inertia is also called second moment of area.

All images are in the public domain.

1. Bend so the top face

(in white) is folding in

on itself.

2. Rotate 90 degrees

and try again.

17

Area moment of inertia

► Below are equations for area moment of inertia for common shapes

𝑟

𝑟𝑜𝑟𝑖

ℎ

𝑏

𝐼 =𝜋

4𝑟4

𝐼 =𝜋

4(𝑟𝑜

4 − 𝑟𝑖4)

𝐼 =1

12𝑏ℎ3

The axis of

rotation is shown

in black here

1818

Quiz

► If the width of a ruler is 3 cm, the thickness is 4 mm, and the length is 30.5 cm,

how much larger is the area moment of inertia for option #2 vs option #1?

ℎ

𝑏

𝐼 =1

12𝑏ℎ31 2

ℎ =

𝑏 =

𝐼 =

ℎ =

𝑏 =

𝐼 =

19

span length

bending loaddeflection

length

support support

Relationship between load and deflection

20

Relationship between fracture strength and deflection

21

Strength in compression

22

Compression: crushing and buckling

► Samples fail in compression by either crushing or buckling.

All images are in the public domain.

If the sample is short and

squat, it will be crushed.

If the sample is long and

slender, it will buckle

(bend out to the side).

23

Crushing

► A sample will be crushed if the applied force exceeds the material’s compressive

strength.

All images are in the public domain. Data: https://www.engineeringtoolbox.com/compression-tension-strength-d_1352.html 4/9/18.

If the sample is short and

squat, it will be crushed.Material

Compressive

Strength

Ultimate

tensile

strength

psi psi

Brickwork,

best quality2000 300

Granite 19000 700

Limestone 9000 300

Sandstone 9000 300

Slate 14000 500

24

Buckling factors

► A sample will fail by buckling if the load is large enough to cause the sample to

deflect to the side.

► What factors affect a sample’s

resistance to buckling? Play around

with some spaghetti and maybe popsicle

sticks to find out.

► Some factors:

► Material

► Length

► Cross-sectional area

► Shape

► How the ends are secured (are they

allowed to rotate?)

All images are in the public domain.

If the sample is long and

slender, it will buckle

(bend out to the side).

25

Buckling equation

► The maximum load a column can support without buckling is called the Euler

critical buckling load.

► Next we’ll talk about the variable K

26

Buckling end conditions

► There are different ways to secure the ends of the columns.

Image from https://wp.optics.arizona.edu/optomech/wp-content/uploads/sites/53/2016/10/OPTI_222_W18.pdf 4/9/18

K = 2 K = 1 K = 0.7 K = 0.5

27

Materials laboratory

► You can now use your new knowledge to measure material properties for spaghetti

► Ultimate tensile stress

► Young’s modulus

► Also determine how strength depends on length, diameter, force, etc